Download Unit 7: Logarithmic Functions

Warm-up
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break in 4 to 6 sentences.
UNIT 6:
LOGARITHMIC
FUNCTIONS
Exponential Function
• Where b is the base and x is the exponent (or power).
• If b is greater than 1, the function continuously increases
in value as x increases.
• A special property of exponential functions is that the
slope of the function also continuously increases as x
increases.
Logarithmic Function
• Simply a way to express the value of an exponent.
• The base of the logarithm is b.
• The logarithmic function has many real-life applications, in
acoustics, electronics, earthquakes analysis and
population prediction
The Graph of the Exponential Function &
the Inverse Function:
The Graph of the Exponential Function &
The Inverse Function:
Exponential Form & Logarithmic Form
Log is just a way to ask a specific question.
loga(b) asks the question:
"What exponent is required to go from a base of
a in order to reach a value of b?"
Examples: Write in logarithmic form
1. 2  y
x
2. 3  27
3
3.
4 =1024
4.
8=2
5
3
Examples: Write in exponential form
1.
log 5 x  2
2. log 6 216  3
3. log 3 ( x  1)  2
4.
log x 4  y
Product & Addition Property
Examples: Write as a single log expression or
write as the sum of multiple log expressions.
1. log 2  log 6
3.
log 10
2. log 7  log x
4.
log 2 z
Quotient & Subtraction Property
Examples: Write as a single log expression or write
as the difference of multiple log expressions.
12
log
4
1. log 9  log 3
3.
2. log 24  log x
5a
4. log
b
Power & Coefficient Property
Important Rule!!
• When you have a number under the radical we can
rewrite that as a fractional exponent and then follow the
Power & Coefficient Property.
1
2
1
log x = log x = log x
2
Examples: Write the log expression with a leading
coefficient or with an exponent.
1.
2.
log 6
2
log x
y
5. log
3
y
2
3.
a log 5
4.
3 log 8
Summary
Put It All Together
• These properties will most likely all be presented together
in a question!
• They will be given with all having the same base
• You will be asked to expand each of the expressions
using the logarithmic properties
OR
• You will be asked to write each expression as a single
logarithm
Examples: Expand or condense the following
logarithmic expressions
1
1. log10 x  z log10 6
2
2. log 3 a  2 log 3 (b  1)
3. log 2 a 4b 5
9
m
4. log 4 2
n
Common Log
The graphing calculators are written in base 10.
Examples:
1. log10 4
2. log10 6
3. l og10 3
4.
log 17
Remember, if you
don’t see a 10 for
the base it is still a
common base so it
can be plugged
right into the
calculator as is.
Natural Logs
ln e  1
 All of the Log Properties also apply to
natural logs
 Multiplication & Addition
 Division & Subtraction
 Power & Coefficient
Examples: Simplify the following expressions
1.
ln e
2. ln e
x
0.017
3. ln 2  ln 3
4. ln 7  ln 4
Examples: Simplify the following expressions
5.
ln 3
2
6. 5 ln 8
7. ln 34  ln 45
8.
ln x  ln y
2
3
Examples: Solve the equation by using Natural Logs
9. e  3
x
10. e
2x
 34
Homework:
• Complete all the even numbers from the work sheet
provided